Generators of the product of two Schoenflies motion groups

Schoenflies motion is often termed X-motion for conciseness. A set of X-motions with a given direction of its axes of rotations has the algebraic properties of a Lie group for the composition product of rigid-body motions or displacements. The product of two X-subgroups, which is the mathematical model of a serial concatenation of two kinematic chains generating two distinct X-motions, characterizes a noteworthy type of 5-dimensional (5D) displacement set called double Schoenflies motion or X–X motion for brevity. This X–X motion set is a 5D submanifold of the displacement 6D Lie group. Such a motion type includes any spatial translation (3T) and any two sequential rotations (2R) provided that the axes of rotation are parallel to two fixed independent vectors. This motion set also contains the rotations that are products of the foregoing two rotations. In the paper, some preliminary fundamentals on the 4D X-motion are recalled; the 5D set of X–X motions is emphasized. Then implementing serial arrays of one-dof Reuleaux pairs and hinged parallelograms, we enumerate all serial mechanical generators of X–X motion, which have no redundant internal mobility. Based on the group-theoretic concepts, one can differentiate two families of irreducible representations of an X–X motion. One family is realized by twenty-one open chains including the doubly planar motion generators as special cases. The other is generally classified into eight major categories in which one hundred and six distinct open chains generating X–X motion are revealed and nineteen more ones having at least one parallelogram are derived from them. Meanwhile, these kinematic chains are graphically displayed for a possible use in the structural synthesis of parallel manipulators.     

Three-dimensional static and dynamic stiffness analyses of the cable driven parallel robot with non-negligible cable mass and elasticity

This study investigates the three-dimensional static and dynamic stiffness analyses of the cable driven parallel robot by considering cable mass, elasticity, and mass of end-effector. According to these models, optimization of cable tensions and cable lengths using fminimax solver is performed to determine the static stiffness. Static and dynamic stiffness of the cables are obtained with simulations. Results show that analyses in three-dimension are very important to measure the actual performance of the cable driven parallel robot. This demonstrates potential for general applicability and motion of the cable driven parallel robot.     

The Kinematics and the Full Minimal Dynamic Model of a 6-DOF Parallel Robot Manipulator

In this paper we present a particular architecture of parallel robots which has six-degrees-of-freedom (6-DOF) with only three limbs. The particular properties of the geometric and kinematic models with respect to that of a classical parallel robot are presented. We show that inverse problems have an analytical solution. However, to solve the direct problems, an efficient numerical procedure which needs to inverse only a 3 × 3 passive Jacobian matrix is proposed. In a second step, dynamic equations are derived using the Lagrangian formalism where the joint variables are passive and active joint coordinates. Based on the geometrical properties of the robot, the equations of motion are derived in terms of only nine coordinates related by three kinematic constraints instead of 18 joint coordinates. The computational cost of the dynamic model obtained is reduced by using a minimum set of base inertial parameters.     

A unified solution to swing-up control for n-link planar robot with single passive joint based on virtual composite links and passivity

This paper concerns the swing-up control of an n-link revolute planar robot with any one of the joints being passive. The goal is to design and analyze a swing-up controller that can bring the robot into any arbitrarily small neighborhood of the upright equilibrium point, at which all the links are in the upright position. We present a unified solution based on the notion of virtual composite link (VCL), which is a virtual link made up of one or more active links. By using the angles of two series of VCLs separated by the passive joint and using the total mechanical energy of the robot, we design a swing-up controller and analyze the global motion of the robot under the controller. The main new results of this paper are: (1) we obtain a lower bound for each control gain related to the angle of each VCL such that the closed-loop system has only one undesired equilibrium point in addition to the upright equilibrium point, and we present an original proof of the conditions on the control gains for a class of n-link underactuation-degree-one planar robots with an active first joint; (2) we provide a bigger control gain region for achieving the control objective than those of previous results on three- and n-link robots with a passive first joint; (3) we validate the theoretical results via numerical simulations on a 4-link robot with the passive joint in each of the four positions. This paper gains an insight into the passivity-based control of underactuated multiple-DOF systems.     

Rayleigh's transformation for thermally buoyant flows

 Rayleigh's classical transformation t=x/U connecting an unsteady parallel boundary flow to the steady plane Blasius-flow is generalized to t=C·x ^c and applied to the case of natural convection in the vicinity of a vertical flat plate. The parameters C and c of this “chronotopic transformation” (CTT) are determined self-consistently. It is shown that the analytic expressions obtained in this way for temperature and for the main component of the steady velocity field reproduce the “numerically exact” patterns to a good accuracy. Surprisingly, the transversal component of the steady flow (which in the unsteady flow is entirely missing) can also be generated by CTT with a remarkable precision. Moreover, the CTT is also able to extrapolate unsteady buoyant flows to steady ones with a good performance, even if these belong to a different set of boundary conditions (e.g. time-dependent vs. coordinate-dependent wall temperatures). Received on 6 May 1999     

绳牵引并联机器人弹性变形对动平台位姿精度的影响Influence of elastic deformation on pose precision of moving platform for wire-driven parallel robot

针对六自由度绳牵引并联机器人在风洞试验中的应用,分析了牵引绳弹性变形对动平台位姿精度的影响,其实质是运动学正解问题。鉴于牵引绳只能受拉力的特点,以及风洞试验的目的,须考虑系统的刚度和绳拉力的优化。基于系统运动学和动力学方程,推导了系统刚度矩阵;以提高系统主方向刚度为目标函数,对绳拉力进行了动态优化分布,以求解弹性变形;采用L‐M数值方法进行运动学正解,量化分析了两种不同弹性模量的牵引绳对系统刚度的影响,以及弹性变形引起的动平台位姿误差。研究结果表明,以刚度增强为优化目标,有利于提高系统稳定性;采用弹性模量较大的牵引绳,可以有效提高系统的刚度,减小绳长变形引起的飞行器模型位姿误差,满足风洞试验的精度要求。上述结果可为后续机构的改善和系统高精度的力位混合控制提供依据和指导。     

Prototype design of a translating parallel robot

The paper describes the mechanical design of a parallel manipulator for motions of pure translation, whose kinematic analysis has shown very good performances such as a large workspace and small overall dimensions of the mobile platform; in particular, the “Cartesian” structure of the machine allowed to obtain constant accuracy and kinematic properties throughout the workspace. The structural design has minimized the mass of the moving links and, by the combined use of FEM and multibody codes, allowed to take into consideration the high stresses coming from inertial forces and to evaluate a-priori the resulting dynamic properties. A physical prototype has just been built in order to validate the developed models and assess the actual robot performances in real operating conditions. The present research has been partially co-funded by the Italian Ministry of Research and University and by the Polytechnic University of Marche under PRIN03 project “Design and prototyping of application-oriented mini-PKM”.     

Treadmill walking of the pneumatic biped Lucy: Walking at different speeds and step-lengths

Actuators with adaptable compliance are gaining interest in the field of legged robotics due to their capability to store motion energy and to exploit the natural dynamics of the system to reduce energy consumption while walking and running. To perform research on compliant actuators we have built the planar biped Lucy. The robot has six actuated joints, the ankle, knee and hip of both legs with each joint powered by two pleated pneumatic artificial muscles in an antagonistic setup. This makes it possible to control both the torque and the stiffness of the joint. Such compliant actuators are used in passive walkers to overcome friction when walking over level ground and to improve stability. Typically, this kind of robots is only designed to walk with a constant walking speed and step-length, determined by the mechanical design of the mechanism and the properties of the ground. In this paper, we show that by an appropriate control, the robot Lucy is able to walk at different speeds and step-lengths and that adding and releasing weights does not affect the stability of the robot. To perform these experiments, an automated treadmill was built Published in Prikladnaya Mekhanika, Vol. 44, No. 7, pp. 134–142, July 2008.     

Kinematic analysis of multi-loop spatial mechanisms: The five-loop case

Multi-loop spatial mechanisms, or parallel mechanisms, are being used to an increasing degree in robotic applications, because they offer some advantages over the open-chain mechanisms. In fact they exhibit high stiffness and low inertia, a fair compliance to the design requirements, and the ability to hold the power actuators in the base. On the other hand, they are often so complex that they can be analyzed only by digital computers, and in general they are designed on the basis of practical solutions rather than theoretical investigations of their structure and kinematics.This paper on the kinematic analysis of multi-loop spatial mechanisms deals in particular with manipulators with five loops and lower pairs. Two approaches are outlined, according to the different backgrounds of the authors: the decomposition method [9], and a differential approach to the non-linear equation set, 19.(6) which encompasses the direct and inverse kinematic problems, as well as the synthesis of the robot.The results are then specialized for mechanisms with the end-effector supported by three spatial parallelograms, like the DELTA-4 robot. For this class of mechanisms an interesting property is also demonstrated, which dramatically reduces the need for computation in the kinematic problems.

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Sommario I meccanismi spaziali in catena chiusa, o meccanismi paralleli, trovano sempre più largo impiego in robotica, dal momento che presentano taluni vantaggi rispetto ai meccanismi in catena aperta: offrono tra l'altro elevata rigidezza e bassa inerzia, flessibilità nel soddisfare le esigenze progettuali e possibilità di collocare gli azionamenti nella base. Spesso tuttavia la loro complessità è tale che si possono analizzare solo col computer, e vengono quindi progettati a partire da soluzioni pratiche, piuttosto che da sistematiche basi teoriche. Questo lavoro illustra metodi generali per lo studio cinematico di meccanismi spaziali in catena chiusa, riferendoli al caso di meccanismi con cinque catene e coppie cinematiche elementari. Gli approcci presentati, che riflettono la diversa formazione degli autori, sono: un metodo di decomposizione [9], e un metodo basato sul calcolo differenziale per lo studio del sistema di equazioni non lineari (6) che, in forma implicita, configura i problemi cinematici diretto e inverso e la sintesi del robot.I risultati vengono quindi applicati a un meccanismo il cui elemento terminale è sostenuto da tre parallelogrammi articolati spaziali, come nel caso del robot DELTA-4. Per questa classe di meccanismi si dimostra anche un'interessante proprietà che riduce drasticamente la quantità di calcoli richiesti dal problema cinematico.

     

Adaptive fuzzy formation control for a swarm of nonholonomic differentially driven vehicles

In this paper, an adaptive fuzzy robust H ^∞ controller is proposed for formation control of a swarm of differential driven vehicles with nonholonomic dynamic models. Artificial potential functions are used to design the formation control input for kinematic model of the robots and matrix manipulations are used to transform the nonholonomic model of each differentially driven vehicle into equivalent holonomic one. The main advantage of the proposed controller is the robustness to input nonlinearity, external disturbances, model uncertainties, and measurement noises, in a formation control of a nonholonomic robotic swarm. Moreover, robust stability proof is given using Lyapunov functions. Finally, simulation results are demonstrated for a swarm formation problem of a group of six unicycles, illustrating the effective attenuation of approximation error and external disturbances, even in the case of robot failure.     

HUMAN-LIKE CONTROL FOR SEMI-PASSIVE BIPEDAL ROBOT WALKING WITH VARIABLE LEG STIFFNESS1)

研究了变刚度半被动双足机器人行走控制问题。采用仿人的行走控制策略,使用变刚度双足弹簧负载倒立摆模型,利用模型自稳定性,在双支撑阶段调整后腿刚度使机器人的能量保持在期望能量附近,在单支撑阶段调整摆动腿落地位置控制质点的高度和前向速度。仿真结果表明：本文采用的控制策略可以实现双足机器人在水平面上的稳定行走,无扰动时可以使机器人实现零输入的被动周期行走,有外部扰动时腿部变刚度控制可使机器人总能量恢复平衡并重新进入稳态行走,控制系统具有鲁棒性。     

Optimization Problems of Controlled Multibody Systems Having Spring-Damper Actuators

The optimal control of the motion of mechanical systems is studied. A characteristic feature of these systems is the presence of passive actuators (springs, dampers, etc.). Energy-optimal control laws and structural parameters of nonlinear spring–damper actuators are determined analytically, which is necessary to impart arbitrary motion to a controllable mechanical system with n degrees of freedom. As an example, a numerical solution is presented for the problem of designing an energy-optimal spring actuator for a robot manipulator of closed kinematic structure     

Kinematics of single-loop mechanisms and serial robot arms: A systematic approach

This paper presents a systematic approach, based on displacement group properties, to the kinematic analysis of spatial linkages with one closed loop and to the solution of the inverse kinematic problem for robot manipulators. By using the proposed approach, a set of kinematic chains can be determined such that a first closure equation with one unknown can be derived directly and explicitly. Then the remaining closure equations are obtained: it is proved that they can be expressed in triangular form. The basic algorithms used to solve these equations in closed form are also presented. For each algorithm, the conditions of applicability, the initial information required, the results, the type and form of equations, and the maximum number of solutions are given. The proposed approach is well suited to the symbolic explicit solution of the inverse kinematic problem of a wide range of robut mechanisms. An example of its application is given.

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Sommario Il lavoro presenta un appreceio sistematico, basato sulle proprietà dei gruppi di spostamento, all'analisi cinematica di posizione di meccanismi spaziali ad una maglia e alla cinematica inversa di robot manipolatori seriali. L'approccio consente di determinare un insieme di catene cinematiche per le quali può essere scritta e risolta direttamente una prima equazione di chiusura in una sola incognita. Viene successivamente dimostrato esaustivamente che, per tali catene, le successive equazioni di chiusura possono essere espresse e risolte in forma triangolare. Inoltre sono presentati gli algoritmi di base utilizzabili per la soluzione del problema posto. Per ciascuno di essi sono dati: le condizioni di applicabilità, l'informazione iniziale richiesta, i risultati ottenuti, il tipo e la forma delle equazioni e il massimo numero di soluzioni possibili. L'approccio presentato è utilizzabile per la soluzione simbolica esplicita, manuale o automatica, di un esteso insieme di meccanismi per robot. Viene dato un esempio di uso del metodo.

     

Drag Reduction of a Circular Cylinder Using an Upstream Rod

Experimental studies on the drag reduction of the circular cylinder were conducted by pressure measurement at a Reynolds number of 82 000 (based on the cylinder diameter). A rod was placed upstream of and parallel to the cylinder to control the flow around the cylinder. The upstream rod can reduce the resultant force of the cylinder at various spacing between the rod and the cylinder for α < 5^∘(α defined as the staggered angle of the rod and the cylinder). For α > 10^∘, the resultant force coefficient has a large value, so the upstream rod cannot reduce the force on the cylinder any more. For α = 0^∘ and d/D = 0.5 (where d and D are the diameter of the rod and the cylinder, respectively), the maximum drag of the cylinder reduces to 2.34% that of the single cylinder. The mechanism of the drag reduction of the cylinder with an upstream rod in tandem was presented by estimating the local contributions to the drag reduction of the pressure variation. In the staggered arrangement, the flow structures have five flow patterns (they are the cavity mode, the wake splitting mode, the wake merge mode, the weak boundary layer interaction mode and the negligible interaction mode) according to the pressure distribution and the hydrogen bubble flow visualization. The half plane upwind of the cylinder can be divided to four regions, from which one can easily estimates the force acting on the circular cylinder with an upstream rod in staggered arrangement.     

Conductance and Noncommutative Dynamical Systems

In this work, we introduce the notion of conductance in the context of Cuntz–Krieger C^∗-algebras. These algebras can be seen as a noncommutative version of topological Markov chains. Conductance is a useful notion in the theory of Markov chains to study the approach of a system to the equilibrium state. Our goal is twofold. On one hand, conductance can be used to measure the complexity of dynamical systems, complementing topological entropy. On the other hand, using C^∗-algebras, we can give a natural framework to analyze the path space of a finite graph associated to a Markov shift.     

Mobility and spatiality of parallel robots revisited via theory of linear transformations

Parallel robots are extensively used for various applications including manipulation, machining, guiding, testing and control. The mechanical architecture of parallel robots is based on parallel mechanisms in which a mobile platform is connected to a reference element by at least two legs. Mobility and spatiality are the main structural and kinematic parameters of a parallel robot. These two parameters are defined via the theory of linear transformation and can be easily determined by inspection using the definitions, properties and theorems introduced in this paper. An analytical method to compute these parameters is also presented just for verification and for a better understanding of their meanings. The new formalism presented in this paper is based on spatiality of an elementary open kinematic chain and relative spatiality between two elements of a closed kinematic chain. As far as we are aware, this paper demonstrates for the first time a new formula for calculation of general (full-cycle) mobility of parallel robots that overcomes the drawbacks of Chebychev–Grübler–Kutzbach's mobility criterion largely used for mobility calculation of multi-loop mechanisms. This new formula is easily applicable to parallel robotic manipulators with elementary or complex legs and mobility calculation does not involve the setting up of instantaneous constraint systems associated to the parallel mechanism.     

Acceleration and singularity analyses of a parallel manipulator with a particular topology

In this work, two essential steps, the kinematics and the singularity analysis, dealing with the design process of a parallel manipulator are investigated by means of the theory of screws. The proposed mechanism for the analysis is a parallel manipulator with three degrees of freedom. A simple and compact expression is derived here for the computation of the reduced acceleration state of the moving platform, w.r.t. the fixed platform, by taking advantage of the properties of reciprocal screws, via the Klein form of the Lie algebra e(3). To this end, the reduced acceleration state of the moving platform is written in screw form through each one of the three actuator limbs of the manipulator. Afterwards, the acceleration analysis is completed by taking into proper account the decoupled motion of the parallel manipulator. Of course, as an intermediate step this contribution also provides the velocity analysis of the parallel manipulator. The expressions obtained via screw theory are compact and can be easily translated into computer codes. A numerical example is provided to demonstrate the efficacy of screw theory to efficiently analyze the kinematics of the chosen parallel manipulator. Finally, the numerical results from the kinematic analysis are compared with results produced with a commercially available dynamic and kinematic simulation program.     

Thermal optimization on micromachined convective accelerometer

 A kind of micromachined convective accelerometer without solid proof mass is numerically and experimentally studied in this paper. The accelerometer consists of a micro heater and two temperature sensors which measure the temperature difference between two symmetrical positions on both sides of the micro heater. The temperature difference is caused by free convection due to acceleration. Thermal optimization on the accelerometer is conducted based on numerical simulation. Three important indexes of the accelerometer, linearity, sensitivity and frequency response are discussed respectively. The results show that linearity relates with the non-dimensional number Gr, only when Gr is in the range from 10⁻² to 10³, good linearity can be achieved. The optimum sensor position for high sensitivity and good linearity is near at x/D=0.3. An increase of heating power or cavity size leads to an increase in the sensitivity. The working media that has small density ρ and large thermal diffusivity α is favorable for fast frequency response, the one having large density ρ and small kinematic viscosity υ will be advantageous for high sensitivity. Experimental tests prove that the optimized convective accelerometer has good linearity, high sensitivity and preferable frequency response. Received on 9 March 2001 / Published online: 29 November 2001     

Homoclinic Shadowing

A new method for rigorously establishing the existence of a transversal homoclinic orbit to a periodic orbit (or a fixed point) of diffeomorphisms in Rⁿ is presented. It is a computer-assisted technique with two main components. First, a global Newton’s method is devised to compute a suitable pseudo (approximate) homoclinic orbit to a pseudo periodic orbit. Then, a homoclinic shadowing theorem, which is proved herein, is invoked to establish the existence of a true transversal homoclinic orbit to a true periodic orbit near these pseudo orbits.     

Kinematostatic modeling of compliant parallel mechanisms

The benefits of compliant mechanisms in terms of precision are not easy to exploit because of the limitations of the existing kinematic models used to analyze them. In practice, compliant mechanisms are more sensitive to external wrenches than conventional mechanisms. In this paper, based on the kinematic constraints and the static equilibrium between the joint coordinates and the external wrenches, a general kinematostatic model of compliant parallel mechanisms is presented. Then, this model is differentiated to provide a quasi-static model that makes it possible to calculate the variation of the pose as a linear function of the motion of the actuators and the variation of the external loads through two new matrices: the quasi-static Jacobian matrix and the Cartesian compliance matrix that give a simple and meaningful formulation of the model of the mechanism.